\(\eta \) -portal
The same Yukawa interaction \( h^{\prime }_{\alpha i} \bar{L}_\alpha \eta ^c N_i\)  inducing LFV is also involved in \(N_1\)  annihilation. Due to tight constraints on \(h^{\prime }\)  from LFV, \(N_1N_1\)  pair annihilation is suppressed for small value of \(h^{\prime }\) , which leads to an excess of observed relic density. For instance, if we set \(h^{\prime }_{\alpha i}=0.01\) , then the \(\eta \) -portal annihilation cross section
\(\langle \sigma v\rangle \simeq \sum _{\alpha \beta }\frac{|h^{\prime }_{\alpha 1}h^{\prime *}_{\beta 1} |^2 r_1^2(1-2r_1+2r_1^2)}{24\pi M_{N_1}}\langle v^2\rangle \sim 6\times 10^{-33} \text{cm}^3 \text{s}^{-1},\) 
where \(r_1=M_{N_1}^2/(M_\eta ^2+M_{N_1}^2)\) . Therefore, the contribution of \(\eta \) -portal annihilation is less than \(10^{-7}\)  in our consideration. One pathway to overcame this confliction is considering hierarchal Yukawa structure or co-annihilation mechanism [1]}, [2]}. Besides, it is also possible to employ the freeze-in production of \(N_1\) , which requires tiny Yukawa coupling \(h^{\prime }_{\alpha 1}\sim 10^{-10}\)  [3]}.
\(H\) -portal
The Higgs portal interactions \(N_1N_1H(h)\)  can mediate \(s\) -channel \(N_1N_1\)  annihilation [4]}, [5]}. Correct relic density can be easily obtained when \(M_{N_1}\sim M_{H}/2\)  for EW scale \(M_H\)  [6]}. And the LFV constraints are escaped by assuming small \(h^{\prime }\lesssim 0.01\) . At the meantime, the Higgs portal interactions also induce spin-independent DM-nucleon scattering, which might be probed by DM direct detection experiments.
\(Z^{\prime }\) -portal
Although only \(N_{R3}\)  is charged under the \(U(1)_{B-3L_\mu }\) , mixings of singlet fermions \(N_R\)  also induce an effective coupling of \(N_1\gamma ^\mu \gamma _5 N_1 Z^{\prime }_\mu \) . The \(Z^{\prime }\) -portal interaction can also mediate \(N_1N_1\)  annihilation [7]}, [8]}. Meanwhile, direct LHC search for the dilepton signature \(pp\rightarrow Z^{\prime }\rightarrow \ell ^+\ell ^-\)  requires that \(Z^{\prime }\)  should be best above TeV scale [9]}, [10]}. Due to the Majorana nature of \(N_1\) , the DM-nucleon scattering cross section mediated by \(Z^{\prime }\)  is suppressed by \(v^2_\text{rel}\sim 10^{-6}\)  [11]}.

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